Powerful harmonic charging in a quantum battery

We consider a harmonic charging field as an energy charger for the quantum battery, which consists of an ensemble of two-level atoms. The charging of noninteracting atoms is completely fulfilled, which exhibits a substantial improvement over previous static charging fields. Involving the repulsive interactions of atoms, the fully charging is achieved with shorter charged period over the noninteracting case, yielding an advantage for the charging. Excluding the charging field, a quantum phase transition is induced by the attractive atom-atom interactions, and the interacting atoms become degenerate in the ground state. We find that the degenerate states play a negative role in the charging due to the gapless energies. The atoms with strong attractive interactions can not be charged completely, which is accompanied by a drop of the maximum stored energy.

Figure 1

(a) Charging protocol of $N$ two-level atoms as the QB. At time $t=0$ each atom is in the ground state $|g\rangle$. At the period $T$, the QB is fully charged, and the final state of atoms is ${|e\rangle }^{\otimes N}$ . (b)The QB is charged with a harmonic driving field $Acos\left(\omega t\right)$. During the charging time $0<t<T$, the QB interacts with a harmonic driving field $Acos\left(\omega t\right)$. Finally, the interaction is switched off at the end of charging period $T$. (c) The intrinsic infinite-range interactions between arbitrary two atoms include the repulsive ($g>0$) and attractive ($g<0$) coupling.

Figure 2

Stored energy $E_1\left(T\right)/\mathrm{\Delta }$ in the single-atom battery as a function of charging period $T$ for driving amplitude $A=0.5$ (a), $A=1$ (b), and $A=1.5$ (c). The QB couples to the harmonic charger $Acos\left(\omega t\right)$ (red sold line) and the static charger $A$ (black dashed line), respectively. The analytical results of $E_1\left(T\right)/\mathrm{\Delta }$ in Eq. (13) for the harmonic charging filed are shown in the red circle.

Figure 3

Maximum stored energy $E_{max}/\mathrm{\Delta }$ (red circles) in the single-atom battery as a function of the driving frequency $\omega$ and the amplitude $A$ of the harmonic charger $Acos\left(\omega t\right)$. The contour projection displays the optimal frequency ${\omega }_{max}$ (solid black line) dependent on $A$.

Figure 4

$\langle S_z\rangle /(N/2)$ as a function of the repulsive and attractive coupling strength $\lambda$ for $N=200$ atoms. The analytical results of $\langle S_z\rangle /(N/2)$ for the infinite atoms $N\to \infty$ display the quantum phase transition at ${\lambda }_c=-1$ in the attractive interactions case (black dashed line). The inset shows the scaled energy $e/(N/2)$ for the ground state(green solid line) and the first-excited state energy (blue dashed line) dependent on $\lambda$.

Figure 5

Stored energy $E_N\left(T\right)$ (in unit of $N\mathrm{\Delta }$) as a function of charging period $T$ for different $N$ with interatomic coupling strength $\lambda =1.2$ (a), $\lambda =0.5$ (b), $\lambda =-0.5$ (c), and $\lambda =-1.2$ (d). The driving amplitude is $A=1$.

Figure 6

Maximum stored energy in the battery $E_{N,max}/\left(N\mathrm{\Delta }\right)$ and the optimal charging period $T_{max}$ as a function of atoms number $N$ for interatomic coupling strength $\lambda =\pm 0.5$ (open circle) (a), (c) and $\lambda =\pm 1.2$ (solid circle) (b), (d), respectively. For $\lambda =-1.2$, the inset shows $E_{max}/\left(N\mathrm{\Delta }\right)$ versus $N$ on a log-log scale, showing the slope of the scaling line 0.0973 (b).

Figure 7

(a) Maximum stored energy $E_{max}$ (in unit of $N\mathrm{\Delta }$) and (b) the optimal charging period $T_{max}$ dependent on the interatomic coupling strength $\lambda$ for $N=140$ atoms.